When Trees Collide: An Approximation Algorithm for theGeneralized Steiner Problem on Networks
SIAM Journal on Computing
A General Approximation Technique for Constrained Forest Problems
SIAM Journal on Computing
The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
On-line algorithms for Steiner tree problems (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Provably good routing tree construction with multi-port terminals
Proceedings of the 1997 international symposium on Physical design
Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Improved performance of the greedy algorithm for partial cover
Information Processing Letters
Polynomial time approximation schemes for dense instances of NP -hard problems
Journal of Computer and System Sciences
Approximation algorithms for directed Steiner problems
Journal of Algorithms
Greedily finding a dense subgraph
Journal of Algorithms
A polylogarithmic approximation algorithm for the group Steiner tree problem
Journal of Algorithms
New approaches to covering and packing problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Approximating min-sum k-clustering in metric spaces
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Approximation algorithms for maximization problems arising in graph partitioning
Journal of Algorithms
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Lectures on Discrete Geometry
A Constant-Factor Approximation Algorithm for the Multicommodity
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Beyond Steiner's Problem: A VLSI Oriented Generalization
WG '89 Proceedings of the 15th International Workshop on Graph-Theoretic Concepts in Computer Science
Polylogarithmic inapproximability
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation
Mathematical Programming: Series A and B
A tight bound on approximating arbitrary metrics by tree metrics
Journal of Computer and System Sciences - Special issue: STOC 2003
On-line generalized Steiner problem
Theoretical Computer Science - Special issue: Online algorithms in memoriam, Steve Seiden
Tighter Bounds for Graph Steiner Tree Approximation
SIAM Journal on Discrete Mathematics
On network design problems: fixed cost flows and the covering steiner problem
ACM Transactions on Algorithms (TALG)
The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Simple cost sharing schemes for multicommodity rent-or-buy and stochastic Steiner tree
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
A general approach to online network optimization problems
ACM Transactions on Algorithms (TALG)
Integrality Ratio for Group Steiner Trees and Directed Steiner Trees
SIAM Journal on Computing
Approximate k-Steiner forests via the Lagrangian relaxation technique with internal preprocessing
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for node-weighted buy-at-bulk network design
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for prize collecting forest problems with submodular penalty functions
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Set connectivity problems in undirected graphs and the directed Steiner network problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Tail bounds for occupancy and the satisfiability threshold conjecture
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Maximum Gradient Embeddings and Monotone Clustering
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Improved approximating algorithms for Directed Steiner Forest
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A greedy approximation algorithm for the group Steiner problem
Discrete Applied Mathematics
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Stochastic steiner trees without a root
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Matching based augmentations for approximating connectivity problems
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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An instance of the k -generalized connectivity problem consists of an undirected graph G=(V,E), whose edges are associated with non-negative costs, and a collection ${\mathcal{D}}=\{(S_{1},T_{1}),\ldots,(S_{d},T_{d})\}$ of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph 驴驴G connects a demand (S i ,T i ) when it contains a path with one endpoint in S i and the other in T i . Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in ${\mathcal{D}}$ .Alon, Awerbuch, Azar, Buchbinder and Naor (SODA '04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiple-choice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log驴2 nlog驴2 d) approximation due to Chekuri, Even, Gupta and Segev (SODA '08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA '06) related its inapproximability to that of dense k -subgraph.In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n 2/3驴polylog(n,k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.